Eur. Phys. J. Plus (2022) 137: 654
https://doi.org/10.1140/epjp/s13360-022-02839-9

Regular Article

A trace formula, perturbation determinant and Levinson’s theorem for a class of star graphs

¹ Department of Mathematics, Lahore University of Management Sciences (LUMS), Lahore, Pakistan
² Department of Basic Sciences, School of Civil Engineering, National University of Sciences and Technology (NUST), Islamabad, Pakistan

^a m.usman08@alumni.imperial.ac.uk

Received: 29 August 2021
Accepted: 15 May 2022
Published online: 3 June 2022

Abstract

We consider Schrödinger operators on quantum star graphs with properly connecting permutation invariant conditions at the central vertex, which, among others, include $\delta$ and ${\delta }^{\prime }$ conditions. We derive trace formulas for the difference of resolvents of perturbed and unperturbed Schrödinger operators. These formulas lead to explicit expressions for the perturbation determinants and the spectral shift function. Under certain integrability conditions on the potential function, the perturbation determinant along with its low energy asymptotics leads to an analog of Levinson’s formula.